Intersection of Spectrum for Two Sturm-Liouville Problems with Periodic Boundary Conditions
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摘要: 为研究具有周期边界条件的两个Sturm-Liouville (SL)问题的交叉谱个数，构造一个二维向量SL问题使两个一维SL问题的谱集与该二维向量SL问题的谱集相同，计算出二维SL问题的二重特征值的一个上界MQ，得出二维SL问题的大于MQ的特征值都是单特征值，且只有有限个非单特征值；利用一维SL问题与二维向量SL问题谱集之间的关系，得出具有周期边界条件的两个一维SL问题交叉谱(相同特征值)的个数是有限的，得到具有周期边界条件的一维SL问题的二重特征值个数也是有限的，同时计算出最大二重特征值的上界估计。
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关键词:
- 向量Sturm-Liouville问题 /
- 特征值 /
- 谱 /
- 重数 /
- 势函数
Abstract: In order to study the intersection of the spectra for two Sturm-Liouville (SL) problems with periodic boundary conditions (BCs), a two-dimensional vectorial SL problem is constructed. In this question the spectral sets of two one-dimensional SL problems are equal to the spectral sets of the two-dimensional vectorial SL problem. Then an upper bound MQ of the double eigenvalues of the two-dimensional SL problem is calculated. It is concluded that the eigenvalues greater than MQ of two-dimensional SL problem are single eigenvalues, and there are only a limited number of non single eigenvalues. Using the relationship between the spectral set of one-dimensional SL problem and two-dimensional vectorial SL problem, it is obtained that the number of cross spectra (same eigenvalues) of two one-dimensional SL problems with periodic BCs is limited, and the number of double eigen‐ values of one-dimensional SL problem with periodic BCs is also limited. At the same time, the upper bound esti‐ mation of the maximum double eigenvalues is calculated.-
Keywords:
- vectorial Sturm-Liouville problem /
- eigenvalue /
- spectrum /
- multiplicity /
- potential function
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